The Horvitz–Thompson total and mean are design-unbiased under positive inclusion probabilities

For every $i$, ${\pi }_i>0$ makes the displayed denominator nonzero, so the formula defines a real-valued map on ${\mathcal{X}}_d$. Its domain has the full power-set sigma-algebra, so the preimage of every Borel set is measurable by [F8]; its target is the total estimand from [F1]. Thus [F2] and [F3] make ${\widehat{T}}_{HT}$ an estimator. For each fixed $y$, it has finite support and finite real values, hence is integrable under $P_y^d$; division by the positive integer $N$ gives the same properties for ${\widehat{\bar{y}}}_{HT}$. Thus both satisfy the integrability clause in [F9].

For each $i\in U$, let $A_i=\{s\in {\mathcal{S}}_d:i\in s\}$. Then $I_i={\mathbf{1}}_{A_i}$ by [F1] and [F4], so [F5] and [F6] give $E_d[I_i]=P_d(A_i)={\pi }_i$.

On a sample $s$, the observed values satisfy $z_i=y_i$ for $i\in s$; hence ${\widehat{T}}_{HT}(O_y(s))={\sum }_{i\in U}{I}_i(s)y_i\mathrm{/}{\pi }_i$. Applying [F7] to this finite sum and using step 1.2 gives $E_y^d[{\widehat{T}}_{HT}]={\sum }_{i\in U}\frac{y_i}{{\pi }_i}{E}_d[I_i]={\sum }_{i\in U}{y}_i=T(y)$. Together with the integrability in step 1.1, this is unbiasedness by [F9]. No independence identity is used.

Since $N\ge 1$ is fixed, linearity of the finite weighted expectation gives $E_y^d[{\widehat{\bar{y}}}_{HT}]=E_y^d[{\widehat{T}}_{HT}]\mathrm{/}N=T(y)\mathrm{/}N={\bar{y}}_U$ by step 2.1. Together with step 1.1, [F9] makes this the required unbiasedness claim for the mean estimator as well.